Chapter 2.1, Problem 30E

(a)

To determine

Mean of the corrected measurements in centimeters.

Answer to Problem 30E

Mean of the corrected measurements is 7.62 cm.

Explanation of Solution

Given information:

  $\text{1}\text{in}\text{.}=\text{2}\text{.54}\text{cm}\text{.}$

Mean,

  ${\overline{x}}_{\text{in}\text{.}}=3.2\text{inches}$

Standard deviation,

  ${\text{SD}}_{\text{in}\text{.}}=0.1\text{inches}$

For converting every data value from inches to centimeters, we need to multiply every data value by 2.54.

Since the measurement was 0.2 inches too large, we are required to decrease the measurement (or data value) by 0.2 inches for corrected measurements.

Since mean represents the center of the measurements.

When every data value is multiplied by 2.54, then the center of the distribution is also multiplied by 2.54.

Similarly,

When every data value is decreased by 0.2 inches, then the center of the distribution is also decreased by 0.2 inches.

Then

The new mean of the corrected measurements in centimeters is the mean of the uncorrected measurements in inches decreased by 0.2 inches, multiplied by 2.54.

New mean,

  ${\overline{x}}_{\text{cm}\text{.}}=\left({\overline{x}}_{\text{in}\text{.}}-0.2\right)\cdot 2.54=\left(3.2-0.2\right)\cdot 2.54=\left(3\right)\cdot 2.54=7.62\text{cm}$

Thus,

The mean of the corrected measurements in centimeters is 7.62 cm.

(b)

To determine

Standard deviation of the corrected measurements in centimeters.

Answer to Problem 30E

Standard deviation of the corrected measurements is 0.254 cm.

Explanation of Solution

Given information:

  $\text{1}\text{in}\text{.}=\text{2}\text{.54}\text{cm}\text{.}$

Mean,

  ${\overline{x}}_{\text{in}\text{.}}=3.2\text{inches}$

Standard deviation,

  ${\text{SD}}_{\text{in}\text{.}}=0.1\text{inches}$

For converting every data value from inches to centimeters, we need to multiply every data value by 2.54.

Since the measurement was 0.2 inches too large, we are required to decrease the measurement (or data value) by 0.2 inches for corrected measurements.

Since the standard deviation is the measure of variability.

When every data value is multiplied by 2.54, then the variability of the measurements is also multiplied by 2.54.

And

When every data value is decreased by 0.2 inches, then the variability of the measurements will not be affected.

Then

The new standard deviation of the corrected measurements in centimeter is the standard deviation of uncorrected measurements in inches multiplied by 2.54.

New standard deviation,

  ${\text{SD}}_{\text{cm}\text{.}}=2.54\left({\text{SD}}_{\text{in}\text{.}}\right)=2.54\left(0.1\right)=0.254\text{cm}$

Thus,

Standard deviation of the corrected measurements in centimeters is 0.254 cm.